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完了 at the beginning of a sentence nearly always is an exclamation of despair. "完了, 敵人被打敗了！" sounds weird, it sounds like that the speaker don't want the enemy to be defeated.
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fefeDec 26 '11 at 17:28

"没完" does not have the meaning of never-ending. At least I cannot think of a situation it is used in this way. Another phrase means never-ending is "没完没了"
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fefeDec 26 '11 at 17:35

@fefe I guess I was thinking more along the lines of "說話沒完的人" (a motormouth, someone who talks to no end). As for "完了" I guess I was thinking about more in the context of exclamation or statement such as "完了, 完了, 彻底完了" (It's done/finished, it's done/finished, it's completely done/finished).
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KrazerDec 26 '11 at 18:35

"完了, 完了, 彻底完了" is an exclamation which implies there is no way to recover from the current (bad) situation. It can have the translation you give, only in certain contexts. "說話沒完的人" is a good example for "没完" (never-ending) , but "一個没完的梦" is not.
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fefeDec 27 '11 at 1:50

All answers miss an important point: the two words are total two different stuff at all! 完 is a verb while 了 is an auxiliary word.

完 in case of this question is used as a secondary part of a compound verb, it denotes the result of the dominated verb. Such kind of compound verbs are called Resultative Compound Verbs. In most situations, say 写完了报告, 了 is used with the whole compound verb instead of just 完. They are not connected as they seem!

This is not what the OP is asking, I think. All the "了" should be changed into "呢" in your examples. "了" is not used in this way. One can say "看了" "看完了", but never "没看了" "没看完了". And there is a typo in your last example "么吃" -> "没吃".
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fefeDec 27 '11 at 5:32